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DCDS

The paper investigates the existence of global attractor for a strongly damped wave equation with fully supercritical nonlinearities: $ u_{tt}-Δ u- Δu_t+h(u_t)+g(u)=f(x) $. In the case when the nonlinearities $ h(u_t) $ and $ g(u) $ are of fully supercritical growth, which leads to that the weak solutions of the equation lose their uniqueness, by introducing the notion of limit solutions and using the theory on the attractor of the generalized semiflow, we establish the existence of global attractor for the subclass of limit solutions of the equation in natural energy space in the sense of strong topology.

PROC

The paper studies the longtime dynamics for a nonlinear wave
equation arising in elastic
waveguide model： $u_{tt}- \Delta u-\Delta u_{tt}+\Delta^2 u- \Delta
u_t -\Delta g(u)=f(x)$. It proves that the equation possesses in
trajectory phase space a global trajectory attractor
$\mathcal{A}^{tr}$ and the full trajectory of the equation in
$\mathcal{A}^{tr}$ is of backward regularity provided that the
growth exponent of nonlinearity $g(u)$ is supercritical.

DCDS

The paper investigates the well-posedness and longtime dynamics of the semilinear
wave equation with gentle dissipation： $u_{tt}-\triangle u+\gamma(-\triangle)^{\alpha} u_{t}+f(u)=g(x)$, with $\alpha\in(0,1/2)$. The main results are concerned with the relationships among the growth exponent $p$ of nonlinearity $f(u)$ and the well-posedness and longtime behavior of solutions of the equation.
We show that (i) the well-posedness and longtime dynamics of the equation are of characters of parabolic equations as $1 \leq p < p^* \equiv \frac{N + 4\alpha}{(N-2)^+}$; (ii) the subclass $\mathbb{G}$ of limit solutions has a weak global attractor as $p^* \leq p < p^{**}\equiv \frac{N+2}{N-2}\ (N \geq 3)$.

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